Assuming a strong preference rule, coalitional structures are rooted binary trees with an added branch at the root representing N. The arcs represent player–sets. The pay–offs are calculated using bargaining parameters b(S) which distribute the gain at a vertex of a tree. Coalitions can only change the structure by permissible‘move’. S will change trees, i.e. a transition occurs, if a move increases its pay–off. We define a relation between two trees if there exists two sets of intermediate trees such that transitions can occur in both directions. This relation is an equivalence relation. There is a partial ordering on the resulting equivalence classes. Thus an‘equilibriu’ tree(s) exists. Similar classes which are included in the previous classes, occur under an equivalence relation defined on‘reversible’ moves only. These are classified as either ‘linea’ or‘cyclic’. If all the classes are linear, the two sets of classes are identical, so that the equilibrium can be found by considering reversible moves only. The nature of the equilibrium state is investigated, and a ‘mixed’ tree is defined to represent such states. Comparison with previous approaches is made.